To find the exact value of [tex]\(\sin(255^\circ)\)[/tex], we can use the angle subtraction identity for sine. The key is to express [tex]\(255^\circ\)[/tex] in terms of angles for which we know the exact trigonometric values.
First, note that:
[tex]\[ 255^\circ = 270^\circ - 15^\circ \][/tex]
Using the sine subtraction identity:
[tex]\[ \sin(255^\circ) = \sin(270^\circ - 15^\circ) \][/tex]
The sine subtraction identity states:
[tex]\[ \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \][/tex]
Let [tex]\( a = 270^\circ \)[/tex] and [tex]\( b = 15^\circ \)[/tex]:
[tex]\[ \sin(255^\circ) = \sin(270^\circ)\cos(15^\circ) - \cos(270^\circ)\sin(15^\circ) \][/tex]
We know from trigonometric values:
[tex]\[ \sin(270^\circ) = -1 \][/tex]
[tex]\[ \cos(270^\circ) = 0 \][/tex]
And the exact values for [tex]\( \sin(15^\circ) \)[/tex] and [tex]\( \cos(15^\circ) \)[/tex] are:
[tex]\[ \sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4} \][/tex]
[tex]\[ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
Substitute these values into our equation:
[tex]\[ \sin(255^\circ) = (-1) \cdot \cos(15^\circ) - 0 \cdot \sin(15^\circ) \][/tex]
[tex]\[ \sin(255^\circ) = -\cos(15^\circ) \][/tex]
Since:
[tex]\[ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
We get:
[tex]\[ \sin(255^\circ) = -\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) \][/tex]
[tex]\[ \sin(255^\circ) = \frac{-(\sqrt{6} + \sqrt{2})}{4} \][/tex]
[tex]\[ \sin(255^\circ) = \frac{-\sqrt{6} - \sqrt{2}}{4} \][/tex]
Thus, the exact value of [tex]\(\sin(255^\circ)\)[/tex] is:
[tex]\[ \boxed{\frac{-\sqrt{6} - \sqrt{2}}{4}} \][/tex]
